Abelian theorems, limit properties of conjugate distributions, and large deviations for sums of independent random vectors

A class of multidimensional absolutely continuous distributions is considered. Each distribution has a moment generating function, which is finite in a bounded convex set S and generates a family of the so-called conjugate distributions. We focus our attention on the limit distributions for this family when the conjugate parameter tends to the boundary of S. As in the one-dimensional case, each limit distribution is obtained as a corollary of the Abel-type theorem. The results obtained are utilized for establishing a local limit theorem for large deviations of arbitrarily high order.